Chebyshev approximation in the coefficient inverse problem for an algebraic polynomial with a prescribed lowest coefficient

A. P. Loktionov   ; S. G. Emelyanov

Čebyševskij sbornik, Tome 25 (2024) no. 4, pp. 106-119

Résumé

In the article, by means of specially designed approximation grid nodes, the class of polynomials $L_n(z, u)$ of degree $n \geqslant 1$ is determined, which deviate least from zero on the interval $-1\leqslant u\leqslant 1$, equal to zero at $u = -1$. For polynomials $L_n(z, u)$ a connection with Chebyshev polynomials of the first kind is described; the $n$-point Chebyshev alternance was studied; extrema are found; exact expressions for the roots and coordinates of the maximum and minimum points are obtained; the formula of the senior coefficient is derived; a segment is found where the polynomial increases monotonically and tends to $ +\infty $ as $u \rightarrow + \infty$. Specific examples of the Chebyshev alternance of the second, third and fourth order are given. We consider algebraic polynomials of degree n with real coefficients. When processing the input data, a uniform continuous rate of absolute error was used. The influence of input data error on the quality of approximation in the coefficient inverse problem for an algebraic polynomial with a prescribed lowest coefficient is studied. In the problem of minimizing the influence of the input data error, the objective function is described as an absolute condition number of the problem, equal to the value of the Lebesgue function. The graphical material shows the level of increase in the numerical value of the absolute condition number of the problem when the coordinates of the approximation grid nodes deviate from the optimal ones. To minimize the influence of the input data error on the accuracy of calculating the coefficients of the studied algebraic polynomial, the location of the nodes of the approximation grid was specially designed. With the Chebyshev approximation, the connection of the nodes with the alternance points of the polynomials $L_n(z,u)$ by a linear function is obtained.

Détail

Mots-clés : coefficient inverse problem, algebraic polynomial
Keywords: data processing, Chebyshev polynomials, Chebyshev approximation.